>>> The natural numbers that we use are said to be derived>>> from what so called Peano's Axioms.

>> They *can* be; this is not the only possible formal>> foundation for them.

>>> While these axioms (listed below) give a method of>>> building up counting numbers they do not define or>>> construct basic arithmetic operations like addition,>>> subtraction, multiplication, etc or basic comparisons>>> like that of equality.

>> Equality is assumed to be a known relation. The arithmetic>> operations and the linear ordering on the natural numbers>> are defined using the axioms. This is explained, albeit>> briefly, in the Wikipedia article on the Peano axioms: